FRANKFURT, Germany — It was only a few years ago that some economists were arguing that Europe was “decoupling” from its long dependence on trade with the United States, and predicted that the Continent’s future lay with the so-called tiger economies of Asia.

German carmakers, at least, had a different vision of the future.

The recovery in the U.S. auto market, which produced big earnings growth at Chrysler and Ford in their fourth quarter, has also been a boon for Germany’s Big Three — Daimler, BMW and Volkswagen.

Their double-digit increases in U.S. sales last year reflected an overall surge in demand by American buyers for European and, above all, German products.

Well-designed vehicles and machinery, so coveted a Germany specialty that they can often fetch premium prices, were by far the biggest categories of European exports to the United States.

As a result, overall German exports to the U.S. rose 24 percent in October from a year earlier, outpacing the 18 percent growth for eurozone exports to the United States.

In many ways, German success has let German carmakers invest in further success when it comes to the U.S. market. The German companies are cashing in on years of commitment to the U.S., which remained an important market for them even as the global auto industry trained its sights on China.

Volkswagen, for example, has invested $4 billion in the U.S. since 2008, building a factory in Chattanooga, Tenn., that began churning out Passat sedans in 2011.

“Five years ago we reset the clock here in America,” Martin Winterkorn, the chief executive of Volkswagen, said in Detroit last month. “The Passat was made in America for America.”

—Jack Ewing, The New York Times

Boy, 5, is safe, suspect dead after Alabama hostage standoff

MIDLAND CITY, Ala. — An armed man who had been holding a 5-year-old boy hostage in an underground bunker is dead and the boy has been released unharmed, ending a six-day standoff, law enforcement authorities said Monday.

Stephen E. Richardson, a special agent with the FBI, said the child, identified only as Ethan, was rescued here at about 3:12 p.m. The authorities said that Jimmy Lee Dykes, 65, a Vietnam veteran with a deep distrust of the government, fatally shot a school bus driver last Tuesday before grabbing the boy and taking him to the bunker, which he built in his yard.

For days, law enforcement officers tried to negotiate a peaceful end to the standoff, but refused to discuss details of negotiations, except to suggest that the boy, who they said has a form of autism, was safe.

“Within the past 24 hours, negotiations deteriorated and Mr. Dykes was observed holding a gun,” Richardson said. “At this point, FBI agents, fearing the child was in imminent danger, entered the bunker and rescued the child.”

It was not clear how Dykes was killed, nor was it immediately clear how the FBI monitored Dykes’ activities in the bunker.

The authorities declined to answer questions about the details of the rescue, although earlier in the weeklong siege they said they had talked to him through a length of PVC pipe.

After the rescue, Richardson said Ethan, who appeared unharmed, was taken to a local hospital.

People near the scene described hearing two loud booms like explosions in midafternoon.

Residents said they were relieved the standoff was at an end.

“This is exactly what we prayed for,” said Michael Senn, a church pastor. “We’re just a small community in the Bible Belt. What got us through this tragedy was the community pulling together and praying together.”